If not, is there any way to make it hold?
Note: the random variable $x$ is called $σ^2$-sub-Gaussian if $E[e^{tx}]≤e^{t^2σ^2/2}$.

  • 2
    $\begingroup$ You would at least need some assumption about their dependence structure. For instance, take the trivial counterexample $X_1 \sim$ normal$(0, 1)$ and $X_2 = X_1$. $X_1$ and $X_2$ are both sub-Gaussian but their product is $\chi^2$ which is not sub-Gaussian. $\endgroup$
    – dsaxton
    Commented Jul 2, 2015 at 3:56


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